STATEMENT-1 : `~(phArrq)=~phArrq=phArr~q`
and
STATEMENT-2 : `(phArrq)hArrr=p hArr(q hArrr)`

Options

A. Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1
B. Statement-1 is True, Statement-2 is True, Statement-2 is NOT a correct explanation for Statement-1
C. Statement-1 is True, Statement-2 is False
D. Statement-1 is False, Statement-2 is True

Answer

To determine the truth of the statements provided, we will analyze each statement step by step using truth tables.

### Statement 1:
**`~(P ↔ Q) = ~P ↔ Q = P ↔ ~Q`**

1. **Define Variables**:
Let P and Q be the two propositions. We will evaluate the truth values of P, Q, ~P, and ~Q.

2. **Truth Table Setup**:
- Create a truth table with columns for P, Q, ~P, ~Q, P ↔ Q, ~P ↔ Q, and P ↔ ~Q.

| P | Q | ~P | ~Q | P ↔ Q | ~P ↔ Q | P ↔ ~Q |
|-------|-------|-------|-------|-------|---------|---------|
| T | T | F | F | T | F | F |
| T | F | F | T | F | T | T |
| F | T | T | F | F | F | T |
| F | F | T | T | T | T | F |

3. **Evaluate Negation**:
- Now, evaluate the negation of P ↔ Q:
- ~ (P ↔ Q) = F, T, T, F (from the column of P ↔ Q).

4. **Check Equivalence**:
- Compare the values of ~ (P ↔ Q), ~P ↔ Q, and P ↔ ~Q:
- ~ (P ↔ Q) = F, T, T, F
- ~P ↔ Q = F, T, T, F
- P ↔ ~Q = F, T, T, F

5. **Conclusion for Statement 1**:
- Since all three expressions yield the same truth values, Statement 1 is **True**.

### Statement 2:
**`(P ↔ Q) ↔ R = P ↔ (Q ↔ R)`**

1. **Define Variables**:
Let P, Q, and R be the three propositions. We will evaluate their truth values.

2. **Truth Table Setup**:
- Create a truth table with columns for P, Q, R, P ↔ Q, Q ↔ R, and (P ↔ Q) ↔ R, P ↔ (Q ↔ R).

| P | Q | R | P ↔ Q | Q ↔ R | (P ↔ Q) ↔ R | P ↔ (Q ↔ R) |
|-------|-------|-------|-------|-------|--------------|--------------|
| T | T | T | T | T | T | T |
| T | T | F | T | F | F | F |
| T | F | T | F | F | F | F |
| T | F | F | F | T | T | T |
| F | T | T | F | T | T | T |
| F | T | F | F | F | F | F |
| F | F | T | T | F | F | F |
| F | F | F | T | T | T | T |

3. **Check Equivalence**:
- Compare the values of (P ↔ Q) ↔ R and P ↔ (Q ↔ R):
- (P ↔ Q) ↔ R = T, F, F, T, T, F, F, T
- P ↔ (Q ↔ R) = T, F, F, T, T, F, F, T

4. **Conclusion for Statement 2**:
- Since both expressions yield the same truth values, Statement 2 is **True**.

### Final Conclusion:
- Statement 1 is **True**.
- Statement 2 is **True**.